CALCULUS BC
WORKSHEET 1 ON VECTORS
Work the following on notebook paper. Use your calculator on problems 10 and 13c only.
3
dy
.
1. If x  t 2  1 and y  et , find
dx
2. If a particle moves in the xy-plane so that at any time t > 0, its position vector is ln  t 2  5t  , 3t 2 , find its
velocity vector at time t = 2.
3. A particle moves in the xy-plane so that at any time t, its coordinates are given
by x  t 5  1 and y  3t 4  2t 3 . Find its acceleration vector at t = 1.


4. If a particle moves in the xy-plane so that at time t its position vector is sin  3t 
vector at time t 
 2
 , 3t , find the velocity
2

.
2
5. A particle moves on the curve y  ln x so that its x-component has derivative x  t   t  1 for t  0. At
time t = 0, the particle is at the point (1, 0). Find the position of the particle at time t = 1.
6. A particle moves in the xy-plane in such a way that its velocity vector is 1  t , t 3 . If the position vector
at t = 0 is 5, 0 , find the position of the particle at t = 2.
7. A particle moves along the curve xy  10. If x  2 and
dy
 3, what is the value of
dx
?
dt
dt
8. The position of a particle moving in the xy-plane is given by the parametric equations
3
x  t 3  t 2  18t  5 and y  t 3  6t 2  9t  4. For what value(s) of t is the particle at rest?
2
9. A curve C is defined by the parametric equations x  t 3 and y  t 2  5t  2. Write the equation of the line
tangent to the graph of C at the point  8,  4  .
10. A particle moves in the xy-plane so that the position of the particle is given by x  t   5t  3sin t and
y  t    8  t 1  cos t  Find the velocity vector at the time when the particle’s horizontal position is x = 25.
11. The position of a particle at any time t  0 is given by x  t   t 2  3 and y  t  
2
3
t 3.
(a) Find the magnitude of the velocity vector at time t = 5.
(b) Find the total distance traveled by the particle from t = 0 to t = 5.
12. Point P  x, y  moves in the xy-plane in such a way that
dx

1
(c) Find
dy
dx
as a function of x.
dy
 2t for t  0.
dt t  1
dt
(a) Find the coordinates of P in terms of t given that t = 1, x  ln 2 , and y = 0.
(b) Write an equation expressing y in terms of x.
(c) Find the average rate of change of y with respect to x as t varies from 0 to 4.
(d) Find the instantaneous rate of change of y with respect to x when t = 1.
and
13. Consider the curve C given by the parametric equations x  2  3cos t and y  3  2 sin t , for 
dy
(b) Find the equation of the tangent line at the point where t 

2

t
.
4
(c) The curve C intersects the y-axis twice. Approximate the length of the curve between the two yintercepts.
(a) Find
dx
as a function of t.

2
.
Answers to Worksheet 1 on Vectors
3
dy 3t 2et
3tet
1.


dx
2t
2
3
2.
9
, 12
14
3. 20, 24
5
 5 
5.  , ln   
 2 
2
6.  9, 4
7. 
8. t = 3
9. y  4  
11. (a) 2600 or 10 26
(b)
4.  3, 3
6
5
10. 7.008,  2.228


12. (a) ln  t  1 , t 2  1
(c)
13. (a)


3
2
26 2  1
3
1
 x  8
12
(c) t  x  3
(b) y   e x  1  1 or y  e2 x  2e x .
2
16
ln 5
(d) 4
2
cot t
3

2
3 2 
(b) y  3  2   x   2 

3 
2  



(c) 3.756